Journal of Theoretical Biology 355 (2014) 111–116
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Stochastic modeling of a serial killer M.V. Simkin n,1, V.P. Roychowdhury n Department of Electrical Engineering, University of California, Los Angeles, CA 90095-1594, United States
H I G H L I G H T S
Serial killers' inter-murder intervals follow a power law with an exponent of 1.5. We hypothesize that they murder when neuronal excitation exceeds a threshold. We model this neural activity as a branching process. Simulations of our model agree with experimental data.
art ic l e i nf o
a b s t r a c t
Article history: Received 6 September 2013 Received in revised form 26 March 2014 Accepted 31 March 2014 Available online 8 April 2014
We analyze the time pattern of the activity of a serial killer, who during 12 years had murdered 53 people. The plot of the cumulative number of murders as a function of time is of “Devil's staircase” type. The distribution of the intervals between murders (step length) follows a power law with the exponent of 1.4. We propose a model according to which the serial killer commits murders when neuronal excitation in his brain exceeds certain threshold. We model this neural activity as a branching process, which in turn is approximated by a random walk. As the distribution of the random walk return times is a power law with the exponent 1.5, the distribution of the inter-murder intervals is thus explained. We illustrate analytical results by numerical simulation. Time pattern activity data from two other serial killers further substantiate our analysis. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Power laws Neural networks Branching process
1. Introduction Fig. 1 shows a time-plot of the cumulative number of murders committed by Andrei Chikatilo (Krivich and Ol'gin, 1993) during his 12-year activity. It is highly irregular with long time intervals without murder interrupted by jumps, when he murdered many people during a short period. Such a curve is known in mathematics as a “Devil's staircase” (Mandelbrot, 1983). We can characterize the staircase by the distributions of step lengths. Fig. 2 shows such distributions for the staircase of Fig. 1 in log–log coordinates. A linear fit shows that the exponent of the power law of the probability density distribution (in the region of more than 16 days) is 1.4. Recently Osorio et al. (2009) reported a similar power-law distribution (with the exponent 1.5) of the intervals between epileptic seizures. Soon afterward they proposed (Osorio et al., 2010) a self-organized critical model of epileptic seizures. They performed numerical simulations of their model and reproduced a power-law distribution of inter-seizure intervals. Almost simultaneously we proposed a stochastic neural network model of epileptic
n
Corresponding authors. E-mail addresses: [email protected] (M.V. Simkin), [email protected] (V.P. Roychowdhury). 1 Tel.: þ1 415 407 6542. http://dx.doi.org/10.1016/j.jtbi.2014.03.039 0022-5193/& 2014 Elsevier Ltd. All rights reserved.
seizures (Simkin and Roychowdhury, 2010), which was very similar to that of Osorio et al. (2010). Unlike them, however, we solved our model analytically. Here we apply a similar model to explain the distribution of intervals between murders.
2. The model We make a hypothesis that, similar to epileptic seizures, the condition, causing a serial killer to commit murder, arise from the simultaneous firing of large number of neurons in the brain. Our neural net model for epileptics (Simkin and Roychowdhury, 2010) and serial killers is as follows. After a neuron has fired, it cannot fire again for a time interval known as the refractory period. Therefore, the minimum interval between the two subsequent firings of a neuron is the sum of spike duration and refractory period. This interval is few milliseconds and we will use it as our time unit. Consider one particular firing neuron. Its axon connects to synapses of thousands of other neurons. Some of them are almost ready to fire: their membrane potential is close to the firing threshold and the impulse from our neuron will be sufficient to surpass this threshold. These neurons will be firing at the next time step and they can be called “children” of our neuron in the language of the theory of branching processes (Simkin and Roychowdhury, 2011). Since the
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cumulative number of murders
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20
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0 12/22/78
8/13/80
4/5/82
11/26/83
7/18/85
3/10/87
10/30/88
6/22/90
Fig. 1. Chikatilo's staircase shows how the total number of his murders grew with time. The time span begins with his first murder on 12/22/1978 and ends with his arrest on 10/20/1990. The shortest interval between murders was three days and the longest—986 days. The murder dates were determined based on the date on disappearance of the person in question.
number of days between murders 10000
1
simulation
10
100
actual data
10000
simulation
1000
0.1
probability density
number of days between murders
1000
1
100
actual data
0.01
0.001
10 0.0001
1 1
10
100
0.00001
rank Fig. 2. Distribution of step length (intervals between murders) in Zipfian (a) and probability density (b) representations.
number of neurons connected to a given neuron is large and since each firing neuron will independently induce the firing of each of the neurons connected to it with a small probability, the number of firings induced by one firing neuron is binomially distributed with a large number of trials and a small success probability, which can be approximated by a Poisson random variable. In addition to induced firings, some neurons will fire spontaneously. We assume that the number of spontaneously firing neurons at each time step comes from a Poisson distribution with mean p. Let us introduce the following random variables: Xn ¼number of firing neurons at time n. Yn ¼number of spontaneously firing neurons at time n. Zn,j ¼number of firings induced by the jth firing neuron at time n.
Then, the process (Xn) defines a discrete-time Markov chain which we obtain by assuming that the two collections of random variables Yn and Zn,j are collections of independent Poisson random variables with mean p and λ, respectively, and by setting X n þ 1 ¼ Z n;1 þ ⋯ þ Z n ; X n þ Y n þ 1 The above equation can be rewritten as (here E (…) denotes the expectation value) X n þ 1 ¼ EðZ n;1 þ ⋯ þ Z n ; X n Þ þ EðY n þ 1 Þ þ ðZ n;1 þ ⋯ þ Z n ; X n EðZ n;1 þ ⋯ þ Z n ; X n ÞÞ þ Y n þ 1 EðY n þ 1 Þ
ð1Þ
M.V. Simkin, V.P. Roychowdhury / Journal of Theoretical Biology 355 (2014) 111–116
In the limit of large Xn we can apply the Central Limit Theorem and get Z n;1 þ ⋯ þZ n ; X n EðZ n;1 þ ⋯ þ Z n ; X n Þ ) X n1=2 Nð0; λÞ;
ð2Þ
where ) means convergence in distribution and N(0,λ) is normal distribution with mean 0 and variance λ. By substituting Eq. (2) into Eq. (1) and noting that E (Zn,1 þ ⋯ þZn, Xn)¼ λ Xn and E(Yn þ 1)¼p we obtain that in the limit of large Xn X n þ 1 ¼ λX n þp þ X 1=2 n zð0; λÞ;
ð3Þ
where z(0,λ) is a normal random variable with mean 0 and variance λ. We can rewrite Eq. (3) as pffiffiffiffi ΔX ¼ ðλ 1ÞX þ p þ X zð0; λÞ
ð4Þ
The number of firing neurons, X, p performs a random walk, with ffiffiffiffi the size of the step proportional to X p. ffiffiffiWe ffi can simplify Eq. (4) by changing variable from X to x ¼ X . We get (Simkin and
113
Roychowdhury, 2010) ðλ 1Þ 1 p 1 1 x þ z Δx ¼ 2 8 2 x 2
ð5Þ
when λ¼ 1, the first term disappears. In the limit of large x, we can neglect the second term, since it is inversely to x. Eq. (5) pffiffiffiproportional ffi reduces to Δx ¼ 1=2z which means that X performs a symmetric Gaussian random walk. A well known problem in random walk theory (Feller, 1957) is that of first return times. That is if the random walk is at point x, after how many steps it will return back to x for the first time. A well known result is that the distribution of first return times follows a power law with an exponent of 3/2 (Feller, 1957). We assume that the killer commits murder when the number of firing neurons reaches certain threshold. Then the distribution of first returns into murder zone (inter-murder intervals) is the same as the distribution of random walk's return times. The model needs to be a bit more complex. We cannot expect that the killer commits murder right at the moment when neural excitation reaches a certain threshold. He needs time to plan and
60
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50
40
30
20
10
0 0
4
2
6 time (in years)
8
10
12
Fig. 3. Results of numerical simulation of the stochastic serial killer model. The distribution of step length is shown in Fig. 2.
0.04
0.035
0.03
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murder probability
theory average rate
0.025
0.02
0.015
0.01
0.005
0 0
200
400
600
800
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1200
days since last murder Fig. 4. Daily murder probability as a function of the number of days passed since the last murder. The average murder rate is the total number of murders committed by Chikatilo divided by the length of the period during which he committed those murders.
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50
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0 9/19/2000
4/7/2001
10/24/2001
5/12/2002
11/28/2002
6/16/2003
Fig. 5. Cumulative number of murders committed by Yang Xinhai.
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0 7/16/94
9/4/94
10/24/94
12/13/94
2/1/95
3/23/95
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7/1/95
8/20/95
10/9/95
Fig. 6. Cumulative number of murders committed by Moses Sithole.
prepare his crime. So we assume that he commits a murder after the neural excitation was over the threshold for a certain period d. Another assumption that we make is that a murder exercises a sedative effect on the killer, causing the number of excited neurons to fall to the threshold. If we do not make this last assumption, the neural excitation will be in the murder zone for half of the time.
3. Numerical simulations We made numerical simulations of the above model. We set λ ¼ 1 which corresponds to the critical branching process in neuron firing. This selection is not arbitrary since some experiments (Beggs and Plenz, 2003) suggest that neural circuits operate in critical regime. There are also theoretical reasons to believe that the brain functions in a critical state (Bak, 1999). The system was simulated for 2 1011 time steps. Remember that time step is the sum of firing duration and refractory period. A reasonable estimate for this is two milliseconds. Thus, our simulation run corresponds to about 12 years. The rate of spontaneous firing was set at p ¼ 0:1. We set the intensity threshold at 109 firing neurons. We set the time threshold, d, at 24 hours.
Figs. 2 and 3 show the results of these simulations. They qualitatively agree with the experimental data. The major disagreement is probably that the actual minimum number of days between murders is three, while the simulation produces a dozen inter-murder periods of one day. One could enhance the model by introducing a murder success rate. That is, with certain probability everything goes well for the killer and he is able to commit the murder as he planned. If not, he repeats his attempt the next day. And so on. One could surely obtain a better agreement with experimental data, but this would be achieved by the price of introducing an extra parameter into the model.
4. Discussion To our knowledge, the only previous mathematical study of the time patterns of serial killers is that by Lange (1999). He used a polynomial map to express the (n þ1)th inter-murder interval through nth and (n 1)th intervals. Afterward he used regression to find the coefficients of the polynomial that will maximize the correlation coefficient between the actual time series of intermurder intervals and the time series obtained using the polynomial map. However, Lange did not propose any plausible mechanism to justify his model.
M.V. Simkin, V.P. Roychowdhury / Journal of Theoretical Biology 355 (2014) 111–116
inter-murder interval 0.1
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1000 0.1
0.01
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1000
probability
probability
0.01
0.001
0.001
0.0001
0.0001 Fig. 7. Probability distribution of inter-murder intervals for Yang Xinhai (a) and Moses Sithole (b).
inter-murder interval 0.1
1
10
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1000
10000
y = 1.9314x-1.6201 0.01
probability
The random-walk models have already been used in Psychology to model human behavior (Estes, 1960). However, those models deal with a choice between several opportunities, not with intervals between events. The connection between the present research and those earlier models may be worthy of investigation in the future. An interesting question to ask is how the probability to commit a new murder depends on the time passed since the last murder. Suppose the killer committed his last murder n days ago. From the random-walk approximation, we immediately get that the probability to commit a murder today is equal to ð1=n3=2 Þ= R1 3=2 ¼ 1=ð2nÞ. Fig. 4 shows this curve together with the n dm=m actual data. The actual murder probability on the nth day after previous murder is computed the following way. The number of instances, when there had been no murder for n 1 days is the total number of murders that happened on nth or later day after the previous one. Thus the murder frequency on the nth day is the ratio of the number of murders, which happened exactly on nth day to the number of murders that happened on nth or later day. To estimate the probabilities we average these frequencies over corresponding bins. There is at least a qualitative agreement between theory and observation. In particular, the probability of a new murder is significantly higher than the average murder rate immediately after murder and is significantly lower than the average murder rate when long time has passed since the last murder. Looking at Fig. 2(b) one can notice a cusp near 100-day intermurder interval. This may suggest that there is a characteristic time scale in the distribution. In one of their experiments with rats Osorio et al. (2010) have found a characteristic time scale mixed in with a scale-free behavior in the distribution of inter-seizure intervals. However in our case this is most likely a statistical fluctuation due to smallness of the sample. Murder patterns of other serial killers are similar to that of Chikatilo. Figs. 5 and 6 show the cumulative number of murders committed by Yang Xinhai (〈http://en.wikipedia.org/wiki/ Yang_Xinhai〉) and Moses Sithole (〈http://www.crimeandinvestigation.
0.001
0.0001
0.00001 Fig. 8. Probability distribution of combined inter-murder intervals for Andrei Chikatilo, Yang Xinhai, and Moses Sithole.
co.uk/crime-files/moses-sithole-south-africa-s-worst-serial-killer/time line.html〉). One obvious difference of Yang Xinhai's staircase from Chikatilo's staircase is the higher step heights: Yang Xinhai killed up to five people at a time. The distribution of step length, however, is very similar as one can see from Fig. 7. One noticeable difference with the Chikatilo case is that we see two-day intervals between murders for Yang Xinhai and one-day intervals for Moses Sithole. This, however, does not imply any real difference in murder patterns but is most likely due to chance: no killer committed enough murders to accumulate accurate intervals statistics. This is also obvious from
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how scattered the plots in Figs. 2 and 7 look. We can partially overcome this problem by combining intervals from all three killers. Fig. 8 shows such plot. As one can see, it is far less scattered than the other plots, although a strong deviation from a power law for short inter-murder intervals remains. Acknowledgements This work was supported in part by the NIH grant No. R01 GM105033-01. Computations were performed at the UCLA IDRE Hoffman2 cluster. References Bak, P., 1999. How Nature Works: The Science of Self-organized Criticality. Copernicus, New York. Beggs, J.M., Plenz, D., 2003. Neuronal avalanches in neocortical circuits. J. Neurosci. 23, 11167.
Estes, W.K., 1960. A random-walk model for choice behavior. In: Arrow, K.J., Karlin, S., Suppes, P. (Eds.), Mathematical Methods in the Social Sciences, 1959. Stanford University Press. Feller, W., 1957. An Introduction to Probability Theory and its Applications, vol. 1. John Wiley, New York. Krivich, M., Ol'gin, O., 1993. Comrade Chikatilo: The Psychopathology of Russia's Notorious Serial Killer. Barricade Books, Fort Lee, New Jersey. Lange, R., 1999. A cusp catastrophe approach to the prediction of temporal patterns in the kill dates of individual serial murderers. Nonlinear Dyn. Psychol. Life Sci. 3, 143. Mandelbrot, B.B., 1983. The Fractal Geometry of Nature. Freeman, New York. Osorio, I., Frei, M.G., Sornette, D., Milton, J., 2009. Pharmaco-resistant seizures: selftriggering capacity, scale-free properties and predictability? Eur. J. Neurosci. 30, 1554. Osorio, I., Frei, M.G., Sornette, D., Milton, J., Lai, Y.-C., 2010. Epileptic seizures: quakes of the brain? Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 82, 021919. Simkin, M.V., Roychowdhury, V.P., 2010. An explanation of the distribution of interseizure intervals. Europhys. Lett. 91, 58005. Simkin, M.V., Roychowdhury, V.P., 2011. Re-inventing Willis. Phys. Rep. 502, 1 〈http://arxiv.org/abs/physics/0601192〉. 〈http://en.wikipedia.org/wiki/Yang_Xinhai〉. 〈http://www.crimeandinvestigation.co.uk/crime-files/moses-sithole-south-afri ca-s-worst-serial-killer/timeline.html〉.
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Stochastic modeling of a serial killer M.V. Simkin n,1, V.P. Roychowdhury n Department of Electrical Engineering, University of California, Los Angeles, CA 90095-1594, United States
H I G H L I G H T S
Serial killers' inter-murder intervals follow a power law with an exponent of 1.5. We hypothesize that they murder when neuronal excitation exceeds a threshold. We model this neural activity as a branching process. Simulations of our model agree with experimental data.
art ic l e i nf o
a b s t r a c t
Article history: Received 6 September 2013 Received in revised form 26 March 2014 Accepted 31 March 2014 Available online 8 April 2014
We analyze the time pattern of the activity of a serial killer, who during 12 years had murdered 53 people. The plot of the cumulative number of murders as a function of time is of “Devil's staircase” type. The distribution of the intervals between murders (step length) follows a power law with the exponent of 1.4. We propose a model according to which the serial killer commits murders when neuronal excitation in his brain exceeds certain threshold. We model this neural activity as a branching process, which in turn is approximated by a random walk. As the distribution of the random walk return times is a power law with the exponent 1.5, the distribution of the inter-murder intervals is thus explained. We illustrate analytical results by numerical simulation. Time pattern activity data from two other serial killers further substantiate our analysis. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Power laws Neural networks Branching process
1. Introduction Fig. 1 shows a time-plot of the cumulative number of murders committed by Andrei Chikatilo (Krivich and Ol'gin, 1993) during his 12-year activity. It is highly irregular with long time intervals without murder interrupted by jumps, when he murdered many people during a short period. Such a curve is known in mathematics as a “Devil's staircase” (Mandelbrot, 1983). We can characterize the staircase by the distributions of step lengths. Fig. 2 shows such distributions for the staircase of Fig. 1 in log–log coordinates. A linear fit shows that the exponent of the power law of the probability density distribution (in the region of more than 16 days) is 1.4. Recently Osorio et al. (2009) reported a similar power-law distribution (with the exponent 1.5) of the intervals between epileptic seizures. Soon afterward they proposed (Osorio et al., 2010) a self-organized critical model of epileptic seizures. They performed numerical simulations of their model and reproduced a power-law distribution of inter-seizure intervals. Almost simultaneously we proposed a stochastic neural network model of epileptic
n
Corresponding authors. E-mail addresses: [email protected] (M.V. Simkin), [email protected] (V.P. Roychowdhury). 1 Tel.: þ1 415 407 6542. http://dx.doi.org/10.1016/j.jtbi.2014.03.039 0022-5193/& 2014 Elsevier Ltd. All rights reserved.
seizures (Simkin and Roychowdhury, 2010), which was very similar to that of Osorio et al. (2010). Unlike them, however, we solved our model analytically. Here we apply a similar model to explain the distribution of intervals between murders.
2. The model We make a hypothesis that, similar to epileptic seizures, the condition, causing a serial killer to commit murder, arise from the simultaneous firing of large number of neurons in the brain. Our neural net model for epileptics (Simkin and Roychowdhury, 2010) and serial killers is as follows. After a neuron has fired, it cannot fire again for a time interval known as the refractory period. Therefore, the minimum interval between the two subsequent firings of a neuron is the sum of spike duration and refractory period. This interval is few milliseconds and we will use it as our time unit. Consider one particular firing neuron. Its axon connects to synapses of thousands of other neurons. Some of them are almost ready to fire: their membrane potential is close to the firing threshold and the impulse from our neuron will be sufficient to surpass this threshold. These neurons will be firing at the next time step and they can be called “children” of our neuron in the language of the theory of branching processes (Simkin and Roychowdhury, 2011). Since the
112
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cumulative number of murders
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20
10
0 12/22/78
8/13/80
4/5/82
11/26/83
7/18/85
3/10/87
10/30/88
6/22/90
Fig. 1. Chikatilo's staircase shows how the total number of his murders grew with time. The time span begins with his first murder on 12/22/1978 and ends with his arrest on 10/20/1990. The shortest interval between murders was three days and the longest—986 days. The murder dates were determined based on the date on disappearance of the person in question.
number of days between murders 10000
1
simulation
10
100
actual data
10000
simulation
1000
0.1
probability density
number of days between murders
1000
1
100
actual data
0.01
0.001
10 0.0001
1 1
10
100
0.00001
rank Fig. 2. Distribution of step length (intervals between murders) in Zipfian (a) and probability density (b) representations.
number of neurons connected to a given neuron is large and since each firing neuron will independently induce the firing of each of the neurons connected to it with a small probability, the number of firings induced by one firing neuron is binomially distributed with a large number of trials and a small success probability, which can be approximated by a Poisson random variable. In addition to induced firings, some neurons will fire spontaneously. We assume that the number of spontaneously firing neurons at each time step comes from a Poisson distribution with mean p. Let us introduce the following random variables: Xn ¼number of firing neurons at time n. Yn ¼number of spontaneously firing neurons at time n. Zn,j ¼number of firings induced by the jth firing neuron at time n.
Then, the process (Xn) defines a discrete-time Markov chain which we obtain by assuming that the two collections of random variables Yn and Zn,j are collections of independent Poisson random variables with mean p and λ, respectively, and by setting X n þ 1 ¼ Z n;1 þ ⋯ þ Z n ; X n þ Y n þ 1 The above equation can be rewritten as (here E (…) denotes the expectation value) X n þ 1 ¼ EðZ n;1 þ ⋯ þ Z n ; X n Þ þ EðY n þ 1 Þ þ ðZ n;1 þ ⋯ þ Z n ; X n EðZ n;1 þ ⋯ þ Z n ; X n ÞÞ þ Y n þ 1 EðY n þ 1 Þ
ð1Þ
M.V. Simkin, V.P. Roychowdhury / Journal of Theoretical Biology 355 (2014) 111–116
In the limit of large Xn we can apply the Central Limit Theorem and get Z n;1 þ ⋯ þZ n ; X n EðZ n;1 þ ⋯ þ Z n ; X n Þ ) X n1=2 Nð0; λÞ;
ð2Þ
where ) means convergence in distribution and N(0,λ) is normal distribution with mean 0 and variance λ. By substituting Eq. (2) into Eq. (1) and noting that E (Zn,1 þ ⋯ þZn, Xn)¼ λ Xn and E(Yn þ 1)¼p we obtain that in the limit of large Xn X n þ 1 ¼ λX n þp þ X 1=2 n zð0; λÞ;
ð3Þ
where z(0,λ) is a normal random variable with mean 0 and variance λ. We can rewrite Eq. (3) as pffiffiffiffi ΔX ¼ ðλ 1ÞX þ p þ X zð0; λÞ
ð4Þ
The number of firing neurons, X, p performs a random walk, with ffiffiffiffi the size of the step proportional to X p. ffiffiffiWe ffi can simplify Eq. (4) by changing variable from X to x ¼ X . We get (Simkin and
113
Roychowdhury, 2010) ðλ 1Þ 1 p 1 1 x þ z Δx ¼ 2 8 2 x 2
ð5Þ
when λ¼ 1, the first term disappears. In the limit of large x, we can neglect the second term, since it is inversely to x. Eq. (5) pffiffiffiproportional ffi reduces to Δx ¼ 1=2z which means that X performs a symmetric Gaussian random walk. A well known problem in random walk theory (Feller, 1957) is that of first return times. That is if the random walk is at point x, after how many steps it will return back to x for the first time. A well known result is that the distribution of first return times follows a power law with an exponent of 3/2 (Feller, 1957). We assume that the killer commits murder when the number of firing neurons reaches certain threshold. Then the distribution of first returns into murder zone (inter-murder intervals) is the same as the distribution of random walk's return times. The model needs to be a bit more complex. We cannot expect that the killer commits murder right at the moment when neural excitation reaches a certain threshold. He needs time to plan and
60
cumulative number of murders
50
40
30
20
10
0 0
4
2
6 time (in years)
8
10
12
Fig. 3. Results of numerical simulation of the stochastic serial killer model. The distribution of step length is shown in Fig. 2.
0.04
0.035
0.03
actual data
murder probability
theory average rate
0.025
0.02
0.015
0.01
0.005
0 0
200
400
600
800
1000
1200
days since last murder Fig. 4. Daily murder probability as a function of the number of days passed since the last murder. The average murder rate is the total number of murders committed by Chikatilo divided by the length of the period during which he committed those murders.
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cumulative number ofmurders
60
50
40
30
20
10
0 9/19/2000
4/7/2001
10/24/2001
5/12/2002
11/28/2002
6/16/2003
Fig. 5. Cumulative number of murders committed by Yang Xinhai.
40
cumulative number of murders
35
30
25
20
15
10
5
0 7/16/94
9/4/94
10/24/94
12/13/94
2/1/95
3/23/95
5/12/95
7/1/95
8/20/95
10/9/95
Fig. 6. Cumulative number of murders committed by Moses Sithole.
prepare his crime. So we assume that he commits a murder after the neural excitation was over the threshold for a certain period d. Another assumption that we make is that a murder exercises a sedative effect on the killer, causing the number of excited neurons to fall to the threshold. If we do not make this last assumption, the neural excitation will be in the murder zone for half of the time.
3. Numerical simulations We made numerical simulations of the above model. We set λ ¼ 1 which corresponds to the critical branching process in neuron firing. This selection is not arbitrary since some experiments (Beggs and Plenz, 2003) suggest that neural circuits operate in critical regime. There are also theoretical reasons to believe that the brain functions in a critical state (Bak, 1999). The system was simulated for 2 1011 time steps. Remember that time step is the sum of firing duration and refractory period. A reasonable estimate for this is two milliseconds. Thus, our simulation run corresponds to about 12 years. The rate of spontaneous firing was set at p ¼ 0:1. We set the intensity threshold at 109 firing neurons. We set the time threshold, d, at 24 hours.
Figs. 2 and 3 show the results of these simulations. They qualitatively agree with the experimental data. The major disagreement is probably that the actual minimum number of days between murders is three, while the simulation produces a dozen inter-murder periods of one day. One could enhance the model by introducing a murder success rate. That is, with certain probability everything goes well for the killer and he is able to commit the murder as he planned. If not, he repeats his attempt the next day. And so on. One could surely obtain a better agreement with experimental data, but this would be achieved by the price of introducing an extra parameter into the model.
4. Discussion To our knowledge, the only previous mathematical study of the time patterns of serial killers is that by Lange (1999). He used a polynomial map to express the (n þ1)th inter-murder interval through nth and (n 1)th intervals. Afterward he used regression to find the coefficients of the polynomial that will maximize the correlation coefficient between the actual time series of intermurder intervals and the time series obtained using the polynomial map. However, Lange did not propose any plausible mechanism to justify his model.
M.V. Simkin, V.P. Roychowdhury / Journal of Theoretical Biology 355 (2014) 111–116
inter-murder interval 0.1
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probability
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0.001
0.0001
0.0001 Fig. 7. Probability distribution of inter-murder intervals for Yang Xinhai (a) and Moses Sithole (b).
inter-murder interval 0.1
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y = 1.9314x-1.6201 0.01
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The random-walk models have already been used in Psychology to model human behavior (Estes, 1960). However, those models deal with a choice between several opportunities, not with intervals between events. The connection between the present research and those earlier models may be worthy of investigation in the future. An interesting question to ask is how the probability to commit a new murder depends on the time passed since the last murder. Suppose the killer committed his last murder n days ago. From the random-walk approximation, we immediately get that the probability to commit a murder today is equal to ð1=n3=2 Þ= R1 3=2 ¼ 1=ð2nÞ. Fig. 4 shows this curve together with the n dm=m actual data. The actual murder probability on the nth day after previous murder is computed the following way. The number of instances, when there had been no murder for n 1 days is the total number of murders that happened on nth or later day after the previous one. Thus the murder frequency on the nth day is the ratio of the number of murders, which happened exactly on nth day to the number of murders that happened on nth or later day. To estimate the probabilities we average these frequencies over corresponding bins. There is at least a qualitative agreement between theory and observation. In particular, the probability of a new murder is significantly higher than the average murder rate immediately after murder and is significantly lower than the average murder rate when long time has passed since the last murder. Looking at Fig. 2(b) one can notice a cusp near 100-day intermurder interval. This may suggest that there is a characteristic time scale in the distribution. In one of their experiments with rats Osorio et al. (2010) have found a characteristic time scale mixed in with a scale-free behavior in the distribution of inter-seizure intervals. However in our case this is most likely a statistical fluctuation due to smallness of the sample. Murder patterns of other serial killers are similar to that of Chikatilo. Figs. 5 and 6 show the cumulative number of murders committed by Yang Xinhai (〈http://en.wikipedia.org/wiki/ Yang_Xinhai〉) and Moses Sithole (〈http://www.crimeandinvestigation.
0.001
0.0001
0.00001 Fig. 8. Probability distribution of combined inter-murder intervals for Andrei Chikatilo, Yang Xinhai, and Moses Sithole.
co.uk/crime-files/moses-sithole-south-africa-s-worst-serial-killer/time line.html〉). One obvious difference of Yang Xinhai's staircase from Chikatilo's staircase is the higher step heights: Yang Xinhai killed up to five people at a time. The distribution of step length, however, is very similar as one can see from Fig. 7. One noticeable difference with the Chikatilo case is that we see two-day intervals between murders for Yang Xinhai and one-day intervals for Moses Sithole. This, however, does not imply any real difference in murder patterns but is most likely due to chance: no killer committed enough murders to accumulate accurate intervals statistics. This is also obvious from
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how scattered the plots in Figs. 2 and 7 look. We can partially overcome this problem by combining intervals from all three killers. Fig. 8 shows such plot. As one can see, it is far less scattered than the other plots, although a strong deviation from a power law for short inter-murder intervals remains. Acknowledgements This work was supported in part by the NIH grant No. R01 GM105033-01. Computations were performed at the UCLA IDRE Hoffman2 cluster. References Bak, P., 1999. How Nature Works: The Science of Self-organized Criticality. Copernicus, New York. Beggs, J.M., Plenz, D., 2003. Neuronal avalanches in neocortical circuits. J. Neurosci. 23, 11167.
Estes, W.K., 1960. A random-walk model for choice behavior. In: Arrow, K.J., Karlin, S., Suppes, P. (Eds.), Mathematical Methods in the Social Sciences, 1959. Stanford University Press. Feller, W., 1957. An Introduction to Probability Theory and its Applications, vol. 1. John Wiley, New York. Krivich, M., Ol'gin, O., 1993. Comrade Chikatilo: The Psychopathology of Russia's Notorious Serial Killer. Barricade Books, Fort Lee, New Jersey. Lange, R., 1999. A cusp catastrophe approach to the prediction of temporal patterns in the kill dates of individual serial murderers. Nonlinear Dyn. Psychol. Life Sci. 3, 143. Mandelbrot, B.B., 1983. The Fractal Geometry of Nature. Freeman, New York. Osorio, I., Frei, M.G., Sornette, D., Milton, J., 2009. Pharmaco-resistant seizures: selftriggering capacity, scale-free properties and predictability? Eur. J. Neurosci. 30, 1554. Osorio, I., Frei, M.G., Sornette, D., Milton, J., Lai, Y.-C., 2010. Epileptic seizures: quakes of the brain? Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 82, 021919. Simkin, M.V., Roychowdhury, V.P., 2010. An explanation of the distribution of interseizure intervals. Europhys. Lett. 91, 58005. Simkin, M.V., Roychowdhury, V.P., 2011. Re-inventing Willis. Phys. Rep. 502, 1 〈http://arxiv.org/abs/physics/0601192〉. 〈http://en.wikipedia.org/wiki/Yang_Xinhai〉. 〈http://www.crimeandinvestigation.co.uk/crime-files/moses-sithole-south-afri ca-s-worst-serial-killer/timeline.html〉.
